Shape Loop Space of Pro-discrete Spaces

Document Type : Research Articles

Author

Department of Pure Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran

Abstract

In this paper, considering the

k

th shape loop space

{\overset{ˇ}{Ω}}_{k}^{p} (X, x)

, for an HPol

_{*}

-expansion

p : (X, x) \to ((X_{λ}, x_{λ}), [p_{λ λ^{'}}], Λ)

of a pointed topological space

(X, x)

, first we prove that

{\overset{ˇ}{Ω}}_{k}

commutes with the product under some conditions and then we show that

{\overset{ˇ}{Ω}}_{k}^{p} (X, x) ≅ lim_{\leftarrow} {\overset{ˇ}{Ω}}_{k}^{p} (X_{i}, x_{i})

, for a pro-discrete space

(X, x) = lim_{\leftarrow} (X_{i}, x_{i})

of compact polyhedra. Finally, we conclude that these spaces are metric, second countable and separable.

Keywords